# A2 The Basic Model of Overlapping Generations

## A2.1 The Consumer Within the Model of Overlapping Generations

We investigate the burden of debt in a simple growth model of overlapping generations based on Diamond (1965). This model is useful to investigate the impact of public debt on generations and capital accumulation in a dynamic context. This is a classical model for investigating the role of public debt in a growing economy (see Ihori, 1996).

Consider a closed economy populated by overlapping generations of two- period-lived consumers and firms. In this model, one young and old generation exist at any point in time. The young have no nonhuman wealth, and the lifetime resources of the young correspond to the labor earnings they receive. There may be population growth. Output is durable and may be accumulated as capital. For simplicity, it is assumed that there is no capital depreciation. The physical characteristics of the endowment are important in overlapping-generation economics since durable goods represent an alternative technology for transferring resources through time.

An agent of generation t is born at time t and considers him- or herself “young” in period t and “old” in period t+ 1. The agent dies at time t+ 2. When young, the agent of generation t supplies one unit of labor inelastically and receives wages w_{t}, out of which the agent consumes c_{t}^{1} and saves s_{t} in period t. An agent who saves s_{t }receives (1 + r_{t+1})s_{t} when old, which the agent then spends entirely on consumption, c^{2}+1, in period t+ 1. r_{t} is the rate of interest in period t. There are no bequests, gifts, or other forms of net intergenerational transfers to the young. In each period, two generations are alive, the young and the old.

A member of generation t faces the following budget constraints:

From Eqs. (4.A1) and (4.A2), her or his lifetime budget constraint is given as

Further, her or his lifetime utility function is given as

The utility function u( ) increases in the vector (c^{1}, c^{2}), twice continuously differentiable and strictly quasi-concave. Thus,

Future consumption is a normal good,

where *u*_{12}* = d*^{2}*u/dc*^{1}*dc** ^{2}* and

*u*

_{11}*= d*

^{2}*u/dc*

^{1}*dc*Starvation is avoided in both periods,

^{1}.

A consumer born in period t maximizes her or his lifetime utility (4.A4) subject to the lifetime budget constraint (4.A3) for given w_{t} and r_{t}+_{1}. For simplicity, we assume that the agent is capable of predicting the future course of the economy and that he or she adopts this prediction as her or his expectation of r_{t}+_{1}. Such rational or perfect foresight expectations are independent of past observations and must be self-fulfilling.

Solving this problem for s_{t} yields the optimal saving function of the agent,

where *ds/dw = s _{w}>* 0 follows from the normality of second period consumption. The sign of

*ds/dr = s*is ambiguous since the substitution effect and the income effect offset each other, as explained in Chap. 8.

_{r}